Post navigation

My next article about graph traversal is pre-empted by this breaking news; I’ll pick up that series again soon.

Yesterday morning a coworker forwarded to me an article about a recently patched security hole in Windows, and wondered if I had any thoughts on it. Oh, did I! I read about the exploit with an odd mixture of nostalgia — because I worked on the code in question back in the 1990s — and horror at how long this exploitable bug had been in Windows.

To be clear, I did not write the actual exploitable code; it predates my time at Microsoft. But I was worried while I was reading the article that it might turn out to be my bad! This is the second time that has happened to me, and it is not a pleasant feeling.

If you have examples of “missing restore”-style bugs — security-impacting or not — in real-world code in any language, I would love to see them. Please leave examples in the comments here or on the security blog. Thanks!

It is pretty obvious just from looking at this graph that if you start from the Troll Room and follow the arrows, there are four rooms that you must enter no matter what combination of edges you take: the East-West Passage, the Dam, the Dam Lobby and the Maintenance Room. And moreover, we’ll encounter them in that order. Let’s call these nodes “the inevitable nodes of the Troll Room”. Continue reading →

Last time we made a little recursive algorithm to walk around a directed acyclic graph, or DAG. I made that algorithm a member of the graph class, but did you notice something about it? It did not access any private member of the class. Rather, it only used the public interface to do its work. What information do we need to find all the traversals of a DAG starting from a given node? Only the set of edges which come off that node, which we could determine from a delegate! We could in fact have written this algorithm as the following static method: Continue reading →

Last time we built an immutable structure representing a labeled directed graph, and then built a directed acyclic graph stolen from Zork: (click for larger)

The question at hand is: given a node in a finite DAG, what are all the possible traversals of edges through the graph starting from that node? We know that any traversal of a finite DAG must eventually terminate in a node that has no outgoing edge, and that there are finitely many such traversals. Continue reading →

There are lots of topics in both practical and theoretical computer science that deal with graphs, lots of different kinds of graphs, lots of ways to represent a graph as a data structure, and lots of operations you’d want to perform on such a data structure. Like, “is this node in this graph?” or “is there any path from this node to that node in this graph?” and so on. Programming with graphs is a huge topic. In this series I want to explore a tiny subset of those topics. Specifically:

How can we create an immutable data structure which represents a directed graph with labels on the edges?

Given a directed acyclic graph (DAG), what are all the possible edge traversals through that graph starting from a particular node?

Previously we enumerated all the combinations of a particular size from a sequence by observing that the sequence {50, 60, 70, 80, 90} had combinations of exactly three elements as follows: Continue reading →